Question: How many integer values of $n$ satisfy $-50 < n^3 < 50$?
Answer: We are asked to find the integers whose cubes are between $-50$ and $50$. Since $f(x)=x^3$ is a monotonically increasing function, we can find the least and the greatest integers satisfying the inequality and count the integers between them, inclusive (see graph). Since $3^3=27<50$ and $4^3=64>50$, $n=3$ is the largest solution. Similarly, $n=-3$ is the smallest solution. Therefore, there are $3-(-3)+1=\boxed{7}$ solutions. [asy]size(7cm,8cm,IgnoreAspect); defaultpen(linewidth(0.7)); import graph; real f(real x) { return x*x*x; } draw(graph(f,-4.5,4.5),Arrows(4)); draw((-4.5,50)--(4.5,50),linetype("3 4"),Arrows(4)); draw((-4.5,-50)--(4.5,-50),linetype("3 4"),Arrows(4)); xaxis(-4.5,4.5,Arrows(4)); yaxis(-4.5^3,4.5^3,Arrows(4)); label("$y=50$",(6,50)); label("$y=-50$",(6,-50)); label("$x$",(4.5,0),E); label("$f(x)=x^3$",(0,4.5^3),N); int n; for(n=-3;n<=3;++n) { dot((n,n^3)); } dot((-4,-64),NoFill); dot((4,64),NoFill); label("$(3,27)$",(3,27),W); label("$(4,64)$",(4,64),W);[/asy]